Counting From 52 to 11,108

fluffy once again brings to my attention the work of Inder J Taneja, who got into the Annals of Improbable Research for a fun parlor-game sort of project a couple of months ago. This was for coming up with ways to (most of) the numbers from 44 up to 1,000 using the digits 1 through 9 in order (ascending and descending), in combinations of addition, multiplication, and exponentiation. Taneja got back in Improbable this weekend with a follow-up project, listing the numbers that can be formed all the way out to a pleasant 11,111.

Taneja’s paper, available at, is that rare mathematics paper that you don’t need to be a mathematician to read, although it isn’t going to strike anyone as very enlightening. The ingenuity involved in many of them is impressive, though, and Taneja lists some interesting things such as how many numbers in a given range can’t be made by the digits in ascending or descending order. (Remarkably, to me at least, everything from 1,001 to 2,000 can be done in ascending or descending order.)

What draws my eye are strings of numbers which can’t be formed, and idle curiosity about what the longest impossible string is. A casual glance over them suggests four looks like the longest — 9,931 through 9,934 can’t be made with ascending digits, while there are quite a few strings of impossible numbers descending, such as 11,029 through 11,032. Obviously the longest string of impossible digits is going to keep growing, and can be made arbitrarily large (I’d advise thinking about it, but that spoils my claim that it’s “obviously” so, at least if we take “obviously” to mean what people normally mean by that), but how long it is compared to the range of digits looked at might be interesting.

Another interesting thing to me is strings of numbers that are impossible using ascending and descending digits, such as 11,029 and 11,030, or the triple 11,805, 11,806, and 11807 (flanked, even by an 11,808 that can’t be made with ascending digits, and an 11,804 impossible using descending ones). The count of such doubly-impossible digits rises as the range we look at increases, naturally — there are only seven of them below 7,000, while there are nine of them between 7,001 and 8,000 (see the table on page 161 of the paper) — and how often they turn up, or how long these strings get, are …

Well, it’s absurd to say these are important things to consider. I’d be stunned if any important mathematics came out of looking at this. But it’s fun stuff, and probably quite good to play with if you’re looking for recreational mathematics puzzles. Of course, Dr Taneja’s got a big lead on you.